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Such algebras (“of observables”, or “of operators”) of field theories (from our point of view: in their decategorified form) are considered as living either in Haag-Kastler nets (in the axiomatic Lorentzian formulation of QFT known as AQFT), or, in Euclidean field theory, in sheaves of algebras (see this).

Here I would like to understand this conceptually from the point of view of the QFT of the charged nn-particle. A guiding example will be Witten’s discussion, in hep-th/0504078, of these structures in 2-dimensional conformal field theory. To remove all technical distractions and clearly extract the underlying structure, I shall try to discuss in detail the (n=1)(n=1)-version (quantum mechanics of the point particle), but in a way that smoothly lends itself to categorification.

The following is the result of discussion with Jens Fjelstad.

— From Schrödinger to Heisenberg picture —

– Schrödinger and Segal compared to Heisenberg and Haag-Kastler

In the so-called Schrödinger picture of quantum mechanics, state vectors are propagated in time.

In the so-called Heisenberg picture of quantum mechanics time-dependent observable operators are the objects under consideration.

The Schrödinger picture manifests itself in the conception of quantum field theory that goes back to Segal and says that:

An nn-dimensional quantum field theory is a representation of a category of nncobordisms with extra structure..

This means that vector spaces “of states” are assigned to (n−1)(n-1)-dimensional spaces, and that morphisms between these vector spaces are associated to the nn-dimensional spaces cobounding these.

The Heisenberg picture of quantum mechanics, on the other hand, manifests itself as the other axiomatic formulation of quantum field theory that people have studied, known as algebraic quantum field theory (AQFT) or, sometimes, just local quantum field theory.

This definition says that

A (local, Lorentzian) nn-dimensional quantum field theory on a space MM is something like a (co)sheaf of algebras (“of observables”) on MM.

– The passage from Schrödinger to Heisenberg, 1-functorially –

I describe how the passage from the Schrödinger to the Heisenberg picture looks like in more or less abstract, arrow-theoretic terms. I do this explicitly for the (n=1)(n=1)-particle only, but the idea is that the way this is formulated the passage to higher nn is already visible.

Here is a worldline
ℝ
\mathbb{R}
and here is the category of paths on the worldline:
P1(ℝ).
P_1(\mathbb{R})
\,.
A quantum mechanical system gives rise to a propagation functor
U:P1(ℝ)→Hilb.
U : P_1(\mathbb{R}) \to \mathrm{Hilb}
\,.
In other words, this is a bundle of Hilbert spaces over ℝ\mathbb{R} with connection
Hdt,
H \,dt
\,,
where HH is the Hamiltonian. (Compare the discussion with Bruce.)

We may want to refine this a little: the Hilbert spaces that we assign to each point in time are special, in that they appear as spaces of sections of the charged particle. In particular, this means, in general, that these are pointed Hilbert spaces, in which a particular cyclic vector
v=|0⟩
v = |0\rangle
is singled out, usually called the ground state or the vacuum.

It so happens that there is an alternative way to think about Hilbert spaces with chosen vacuum vectors: namely in terms of C*C^*-algebras with with a chosen state on them.

From every Hilbert space HH with cyclic vector v∈Hv \in H, we get the C*C^*-algebra B(A)B(A) of bounded operators on HH, equipped with the pure normal state defined by a↦⟨v,avrangea \mapsto \langle v, a v\range.

This should in fact be a functor
B:Hilbcyc→Cmathrmsts*.
B : \mathrm{Hilb}_{\mathrm{cyc}} \to C^*_{mathrm{sts}}
\,.

Conversely, given any C*C^*-algebra AA with a pure normal state ϕ\phi on it, the GNS construction provides a Hilbert space HH with a cyclic vector v∈Hv \in H such that A≃B(H)A \simeq B(H) and ϕ(a)=⟨v,avrange\phi(a) = \langle v, a v\range . This should give a functor
GNS:Csts*→Hilbcyc.
\mathrm{GNS} : C^*_{\mathrm{sts}} \to \mathrm{Hilb}_{\mathrm{cyc}}
\,.
Moreover, these two functors should yield an equivalence of categories. (I am very grateful to Masoud Khalkhali for his help with figuring out what the precise formulation of this statement is.)

Given this (presumable) equivalence of the codomain of the Schrödinger picture functors U:P1(X)→HilbcycU : P_1(X) \to \mathrm{Hilb}_{\mathrm{cyc}} with the category Csts*C^*_{\mathrm{sts}}, there is nothing more natural than passing to that equivalent picture by
hitting UU with the functor
B*:[P1(ℝ),Hilb]→[P1(ℝ),C*],
B_* : [P_1(\mathbb{R}),\mathrm{Hilb}]
\to
[P_1(\mathbb{R}),C^*]
\,,
which postcomposes any functor with the functor B:Hilbcyc→Csts*B : \mathrm{Hilb}_{\mathrm{cyc}} \to C^*_{\mathrm{sts}}.

The resulting Heisenberg transportB*U:P1(ℝ)→C*
B_* U : P_1(\mathbb{R})
\to
C^*
is now an algebra bundle with connection over ℝ\mathbb{R}!

(Each fiber should be thought of as the closure of the Weyl algebra characterizing the phase space of the system. )

The connection 1-form now reads
adHdt.
\mathrm{ad}_H \, d t
\,.

To be explicit, assume the QM system is that of a particle propagating on ℝn\mathbb{R}^n (I could consider ℝ1\mathbb{R}^1 for simplicity, but I want to keep us from confusing target space with parameter space).

Then, all the Hilbert space fibers of the original bundle are canonically identified with L2(ℝn)L^2(\mathbb{R}^n).

Accordingly, all the algebra fibers are canonically identified with the Weyl algebra, which I think of, for convenience, as canonically generated infinitesimally from elements xix^i and pip_i with [xi,pj]=iδji[x^i,p_j] = i \delta^i_j.

- sheaves of algebras of observables for n=1n=1 -

Now consider the algebra of flat sections of our algebra bundle over the worldline.

Better yet, consider the sheaf of algebras of flat sections of this bundle.

To every open interval (a,b)⊂ℝ(a,b) \subset \mathbb{R}, this will associate all flat section of B*U|(a,b):P1(a,b)→C*B_* U|_{(a,b)} : P_1(a,b) \to C^*.

But every such flat section is completely determined already by its value over any one codimension one space – i.e. by its value of any one point.

Assume the points 00 and tt are inside our interval.

Over 00, we have the algebra element xix^i. We might call that
xi(0)
x^i(0)
to make explicit that we think of xix^i as sitting in the fiber B*U(0)B_* U(0) and not anywhere else.

This element determines an entire flat section over all of (a,b)(a,b). Over tt, the value of this flat section will be (for the free particle) xi−tmpix^i - \frac{t}{m}p_i, now regarded as an element of B*U(t)B_* U(t). So we should, following the above convention, write this as
xi(t)−tmpi(t).
x^i(t) - \frac{t}{m}p_i(t)
\,.

Let me write
exi(0)∈Γ(B*U)
e_{x^i(0)}
\in
\Gamma(B_* U)
for the flat section uniquely specified by that fact that its value at 00 is xix^i.

Similarly I then write exi(t)−tmpi(t)e_{x^i(t) - \frac{t}{m}p_i(t)} for the flat section uniquely specified by having the value
xi+tmpix^i + \frac{t}{m}p_i over tt.

Since both xi(0)x^i(0) and xi(t)+tmpi(t)x^i(t) + \frac{t}{m}p_i(t) uniquely specify the same flat section of our algebra bundle, we have
exi(0)=exi(t)+tmpi(t).
e_{x^i(0)}
=
e_{x^i(t) + \frac{t}{m}p_i(t)}
\,.
If we were in a more relaxed mood, we might simply write
xi(0)=xi(t)−tmpi(t).
x^i(0)
=
x^i(t) - \frac{t}{m}p_i(t)
\,.

(I know I require a certain tolerance for sophisticated-looking trivialities here. The point is that this exercise is supposed to make life easier as we go up the dimensional ladder.)

Now, sections of a bundle of algebras form an algebra themselves, simply by pointwise multiplication.

In our more relaxed mood, this reproduces the familiar formulas for the operator algebra of fields on the worldline
[xi(0),xj(t)]=tmiδji.
[x^i(0), x^j(t)]
=
\frac{t}{m}
i\delta^i_j
\,.

Bottom line: (sheaves of) operator algebras of fields are (sheaves of) algebras of flat sections of the Heisenberg bundle obtained as the push-forward of the Hilbert bundle obtained in the Schrödinger picture.

- cosheaves and Haag-Kastler nets of algebras of observables for n=1n=1 -

A Haag-Kastler net of observables is defined to be something rather similar to a (pre)cosheaf of algebras. Is that at odds with the sheaves of algebras of observables that we found above (and that are known in Euclidean field theory)?

The sheaves of observables that we have found here are actually rather special. The way they arise, we see that we may just as usefully think of them as cosheaves:

since every flat section is uniquely determined by its value on a codimension one subset (a point in our n=1n=1 example), we have a canonical injection of the algebra AVA_V of flat sections over VV into the algebra AUA_U of flat sections over UU for open sets
V⊂U.
V \subset U
\,.

This is really crucially due to the fact that we are dealing with flat sections of the Heisenberg algebra bundle.

Accordingly, it makes good sense to think of our flat sections as forming a cosheaf of algebras.

Or, alternatively, we may think of them as forming a precosheaf of algebras with the special condition that the corestriction maps are injections.

The latter definition is that appearing in the definition of Haag-Kastler nets of observables.

The full definition of Haag-Kastler nets refers to a Lorentzian structure on parameter space. The presence of such a structure is something I don’t want to restrict attention to. I would like to understand Euclidean and Lorentzian parameter spaces for QFTs (of worldvolume theories of charged particles) on a commen basis.

For that reason, I will talk about Haag-Kastler-like nets of algebras, in the following, whenever I am referring to pre-cosheaves of algebras with injective corestriction maps.

It turns out (at least as far as I can see) that this concept (rather than that of a sheaf of algebras) is the one we need to really perform the next step.

— From Heisenberg to Schrödinger: the Worldvolume Anomaly —

In the practice of quantum field theory, people often find themselves faced with the problem opposite to the one discussed so far: on local patches of parameter space algebras of observables are known, and one tries to reconstruct from that a Hamiltonian that generates the time evolution, globally.

– finding a Hamiltonian –

Suppose on parameter space (in the example that I am concentrating on, this is the real line, ℝ\mathbb{R}, or rather its categorical incarnation, tar=P1(ℝ)\mathrm{tar} = P_1(\mathbb{R})) we have given a Haag-Kastler-like net of algebras
(of observables), i.e. a pre-cosheaf of algebras with injective corestriction maps.

On each patch UU of target space, a local candidate for a Hamiltonian is an element
HU∈AU
H_U \in A_U
of the algebra associated with UU, such that, for V⊂UV \subset U a subset and Vt⊂UV_t \subset U its translation by tt (which makes sense in my example, where n=1n =1, and has an obvious generalizations to n>1n \gt 1) such that
AdeitHU:AV≃AVt,
\mathrm{Ad}_{e^{i t H_U}} : A_V \simeq A_{V_t}
\,,
where this is interpreted in AUA_U using the injective corestriction maps.

In words: an element of the algebra is a candidate for a local Hamiltonian if its action on the algebra can be regarded as the relevant (time-)translation operation.

– gluing the Hamiltonian –

From our discussion of the passage from the Schrödinger to the Heisenberg picture, we know that the Hamiltonian is actually (the single component of) a globally defined connection 1-form on the Hilbert bundle of states, or, equivalently (by its adjoint action) on the Heisenberg bundle of algebras of observables.

In order to reconstruct this situation from our local data we need to

- cover parameter space by open sets

- glue the algebras of observables to sections of an algebra bundle that are flat with respect to the local Hamiltonians

- such that these local Hamiltonians constitute a globally defined connection 1-form on this bundle of algebras.

If this last step fails, i.e. if we have a locally (local on parameter space!) well-defined Hamiltonian which fails to constitute a globally well defined Hamiltonian, we say we have a worldvolume anomaly.

(For the case of a 2-particle being a string with a conformal field theory on its worldvolume (a worldsheet, in this case), this anomaly is precisely (a not so common incarnation of) the famous conformal anomaly).

It is helpful to consider this problem that we are faced with here in the general context of bundles with connection.

For UU any patch, let P1(U)P_1(U) be the category of paths in that. A vector bundle with connection on UU is a parallel transport functor
traU:P1(U)→Vect.
\mathrm{tra}_U : P_1(U) \to \mathrm{Vect}
\,.
A flat (covariantly constant) section ee of this bundle is a morphism of functors
eU:1→traU,
e_U: 1 \to \mathrm{tra}_U
\,,
where 11 denotes the tensor unit in the monoidal category of all such functors.

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